MECH 479: Large Eddy Simulation
W. K. Bushe
University of British Columbia
Department of Mechanical Engineering
Today’s To Do List
• Discuss spatial ﬁltering
• Discuss LES closures
• Discuss LES wall models
What are the diﬀerences between the Turbulent Viscosity models
and Reynolds stress models?
We can deﬁne a spatial ﬁlter in three dimensions like this:
f (xk ) ≡ f (xk − rk )g(xk , rk )drk .
Here, we have to deﬁne what the function g(xk , rk ) is. Once
that is accomplished, we can apply our ﬁlter to the governing
U (xk , t) ≡ U (xk − rk , t)g(xk , rk )drk .
We can now do a Reynolds-like decomposition:
u′(xk , t) ≡ U (xk , t) − U (xk , t).
In this case, though, U (xk , t) is still a random ﬁeld and u′ = 0.
Also, while the derivative in time commutes past the ﬁlter, the
derivative in space only commutes for “homogeneous” ﬁlters.
There are many choices for ﬁlters, with three common ones
• Box (or “top-hat”)
• Sharp spectral
Filtered energy spectrum
The eﬀect of ﬁltering the velocity ﬁeld is to dampen (or elimi-
nate, in the case of the spectral ﬁlter) energy at high wavenum-
bers. That would do this to the energy spectrum:
Given that there is less energy contained at the high wavenum-
bers in a spatially ﬁltered velocity ﬁeld, we can use a much
coarser grid to resolve ﬂuctations in the ﬁltered ﬁeld. In a sense,
this would be like DNS, but only for the motion at the small
wavenumbers (the large-scale motion).
Filtered governing equations
The equations for LES arise from us ﬁltering the governing equa-
So, the ﬁltered equations contain an unclosed term representing
the dissipation of energy that has been ﬁltered away (the energy
“missing” from the energy spectrum). We have to model that
Smagorinski proposed to model the sub-grid scale stresses with
an eddy viscosity model:
∂U j ∂U i
τij = −νT + = −2νT S ij
The eddy viscosity should (obviously) have a length scale asso-
ciated with the ﬁlter dimension ∆. It also has to have a time
scale which comes from us assuming that the ﬁlter length-scale
is in the inertial sub-range; there, the characteristic time should
be a function only of ǫ. How do we get a viscosity from ǫ and
Smagorinski proposed to model the dissipation rate with the sub-
grid dissipation rate:
ǫ ≈ −τij S ij = 2νT S ij S ij .
We then plug this into the expression for the eddy viscosity:
This is an extension of the Smagorinski model that allows us to
compute the Smagorinski “constant” rather than presuming it.
The idea is to deﬁne two diﬀerent ﬁlter lengths: a “grid” ﬁlter
f (xk , t) = f (xk − rk )g(xk , rk )drk
with a characteristic scale of ∆ and a “test” ﬁlter
f (xk , t) = f (xk − rk )g(xk , rk )drk .
Applying the test ﬁlter to the grid ﬁlter would generate this:
g = gg and have a characteristic ﬁlter width of ∆. We still have
τij deﬁned as before (based on the grid ﬁlter). If we ﬁlter the
grid-ﬁltered N-S equations with the test ﬁlter, we get:
We now consider the “resolved turbulent stress”
Lij = uiuj − uiuj = Tij − τij .
Can we calculate that?
If we use the same model for both Tij and τij :
τij = −2C∆ 2S kl S kl S ij
Tij = −2C ∆ 2S kl S kl S ij
then, we can substitute these quantities into the above equation
and solve for C. We actually have six equations for C from that
operation; also, Lij is often zero, so we prefer to calculate an
average C on surfaces in the ﬂow on which it should be constant
(like at the same distance from a wall). This leads to a least-
squares solution for C on the surface.
In actual LES implementations, we would like to avoid deﬁning
a ﬁlter and ﬁltering the data. We could simply deﬁne our grid
so that our ﬁnest resolution is in the intertial sub-range, and
go ahead with an unsteady calculation. The numerical scheme
used to discretize the domain implies a ﬁlter (just because we
don’t resolve the motion that should be present at scales smaller
than the grid). Unfortunately, depending on the scheme you use,
the eﬀective ﬁlter can be diﬀerent for diﬀerent order derivatives.
Another problem is that, with low-order numerics, you get exces-
sive numerical dissipation at the smallest scales. This can end
up exceeding the sub-grid scale dissipation.
Altogether, it seems that explicitly ﬁltering the data leads to
much better results . . .
In LES, walls are bad. For a straight-forward channel ﬂow, the
wall-models for RANS can be modiﬁed to work for LES. It also
works for relatively simple engineering ﬂows, but not for more
complex ones. One practice has been to resolve the ﬂow near
the wall (that is, use DNS in the wall-normal directions). That
leads to us spending up to 50% of the computational eﬀort in
the near-wall region. A lot of eﬀort has been put into ﬁxing this
and some success has been seen recently . . .
Detached Eddy Simulation is a hybrid approach to LES. The
idea is very simple: use the Spalart-Allmaras RANS model for
the eddy viscosity. In their equation for eddy viscosity, there is
a term that is a function of the distance to the wall. If, when
you get far from the wall, you switch to using the local grid
spacing as the “distance”, you get a model that behaves like
the Smagorinski model. That way, you can simulate the ﬂow
near the wall with RANS and the wall models; far from the wall,
you do LES. This has been shown to work very well for airfoils,
including airfoils with ﬂaps, slats, etc.